Found problems: 396
2015 European Mathematical Cup, 1
$A = \{a, b, c\}$ is a set containing three positive integers. Prove that we can find a set $B \subset A$, $B = \{x, y\}$ such that for all odd positive integers $m, n$ we have $$10\mid x^my^n-x^ny^m.$$
[i]Tomi Dimovski[/i]
1998 Poland - Second Round, 5
Let $a_1,a_2,\ldots,a_7, b_1,b_2,\ldots,b_7\geq 0$ be real numbers satisfying $a_i+b_i\le 2$ for all $i=\overline{1,7}$.
Prove that there exist $k\ne m$ such that $|a_k-a_m|+|b_k-b_m|\le 1$.
Thanks for show me the mistake typing
2000 Kurschak Competition, 3
Let $k\ge 0$ be an integer and suppose the integers $a_1,a_2,\dots,a_n$ give at least $2k$ different residues upon division by $(n+k)$. Show that there are some $a_i$ whose sum is divisible by $n+k$.
2004 Canada National Olympiad, 5
Let $ T$ be the set of all positive integer divisors of $ 2004^{100}$. What is the largest possible number of elements of a subset $ S$ of $ T$ such that no element in $ S$ divides any other element in $ S$?
PEN P Problems, 21
Let $A$ be the set of positive integers of the form $a^2 +2b^2$, where $a$ and $b$ are integers and $b \neq 0$. Show that if $p$ is a prime number and $p^2 \in A$, then $p \in A$.
2013 Moldova Team Selection Test, 2
Let $a_n=1+n!(\frac{1}{0!}+\frac{1}{1!}+\frac{1}{2!}+...+\frac{1}{n!})$ for any $n\in \mathbb{Z}^{+}$. Consider $a_n$ points in the plane,no $3$ of them collinear.The segments between any $2$ of them are colored in one of $n$ colors. Prove that among them there exist $3$ points forming a monochromatic triangle.
2024 AMC 10, 12
A group of $100$ students from different countries meet at a mathematics competition. Each student speaks the same number of languages, and, for every pair of students $A$ and $B$, student $A$ speaks some language that student $B$ does not speak, and student $B$ speaks some language that student $A$ does not speak. What is the least possible total number of languages spoken by all the students?
$
\textbf{(A) }9 \qquad
\textbf{(B) }10 \qquad
\textbf{(C) }12 \qquad
\textbf{(D) }51 \qquad
\textbf{(E) }100 \qquad
$
2008 ISI B.Math Entrance Exam, 4
Let $a_1,a_2,...,a_n$ be integers . Show that there exists integers $k$ and $r$ such that the sum
$a_k+a_{k+1}+...+a_{k+r}$
is divisible by $n$ .
2022 Cyprus JBMO TST, 4
Let $A$ be a subset of $\{1, 2, 3, \ldots, 50\}$ with the property: for every $x,y\in A$ with $x\neq y$, it holds that
\[\left| \frac{1}{x}- \frac{1}{y}\right|>\frac{1}{1000}.\]
Determine the largest possible number of elements that the set $A$ can have.
2009 AIME Problems, 13
The terms of the sequence $ (a_i)$ defined by $ a_{n \plus{} 2} \equal{} \frac {a_n \plus{} 2009} {1 \plus{} a_{n \plus{} 1}}$ for $ n \ge 1$ are positive integers. Find the minimum possible value of $ a_1 \plus{} a_2$.
2012 China Team Selection Test, 3
Let $a_1<a_2$ be two given integers. For any integer $n\ge 3$, let $a_n$ be the smallest integer which is larger than $a_{n-1}$ and can be uniquely represented as $a_i+a_j$, where $1\le i<j\le n-1$. Given that there are only a finite number of even numbers in $\{a_n\}$, prove that the sequence $\{a_{n+1}-a_{n}\}$ is eventually periodic, i.e. that there exist positive integers $T,N$ such that for all integers $n>N$, we have
\[a_{T+n+1}-a_{T+n}=a_{n+1}-a_{n}.\]
2012 Online Math Open Problems, 44
Given a set of points in space, a [i]jump[/i] consists of taking two points, $P$ and $Q,$ and replacing $P$ with the reflection of $P$ over $Q$. Find the smallest number $n$ such that for any set of $n$ lattice points in $10$-dimensional-space, it is possible to perform a finite number of jumps so that some two points coincide.
[i]Author: Anderson Wang[/i]
2011 USA Team Selection Test, 9
Determine whether or not there exist two different sets $A,B$, each consisting of at most $2011^2$ positive integers, such that every $x$ with $0 < x < 1$ satisfies the following inequality:
\[\left| \sum_{a \in A} x^a - \sum_{b \in B} x^b \right| < (1-x)^{2011}.\]
2009 Ukraine National Mathematical Olympiad, 4
Let $G$ be a connected graph, with degree of all vertices not less then $m \geq 3$, such that there is no path through all vertices of $G$ being in every vertex exactly once. Find the least possible number of vertices of $G.$
2001 USAMO, 1
Each of eight boxes contains six balls. Each ball has been colored with one of $n$ colors, such that no two balls in the same box are the same color, and no two colors occur together in more than one box. Determine, with justification, the smallest integer $n$ for which this is possible.
1999 Croatia National Olympiad, Problem 4
Given nine positive integers, is it always possible to choose four different numbers $a,b,c,d$ such that $a+b$ and $c+d$ are congruent modulo $20$?
2008 APMO, 2
Students in a class form groups each of which contains exactly three members such that any two distinct groups have at most one member in common. Prove that, when the class size is $ 46$, there is a set of $ 10$ students in which no group is properly contained.
2009 Philippine MO, 3
Each point of a circle is colored either red or blue.
[b](a)[/b] Prove that there always exists an isosceles triangle inscribed in this circle such that all its vertices are colored the same.
[b](b)[/b] Does there always exist an equilateral triangle inscribed in this circle such that all its vertices are colored the same?
1985 IMO Shortlist, 1
Given a set $M$ of $1985$ positive integers, none of which has a prime divisor larger than $26$, prove that the set has four distinct elements whose geometric mean is an integer.
1997 South africa National Olympiad, 6
Six points are connected in pairs by lines, each of which is either red or blue. Every pair of points is joined. Determine whether there must be a closed path having four sides all of the same colour. (A path is closed if it begins and ends at the same point.)
2013 ELMO Shortlist, 3
Let $a_1,a_2,...,a_9$ be nine real numbers, not necessarily distinct, with average $m$. Let $A$ denote the number of triples $1 \le i < j < k \le 9$ for which $a_i + a_j + a_k \ge 3m$. What is the minimum possible value of $A$?
[i]Proposed by Ray Li[/i]
2009 Germany Team Selection Test, 1
In the coordinate plane consider the set $ S$ of all points with integer coordinates. For a positive integer $ k$, two distinct points $A$, $ B\in S$ will be called $ k$-[i]friends[/i] if there is a point $ C\in S$ such that the area of the triangle $ ABC$ is equal to $ k$. A set $ T\subset S$ will be called $ k$-[i]clique[/i] if every two points in $ T$ are $ k$-friends. Find the least positive integer $ k$ for which there exits a $ k$-clique with more than 200 elements.
[i]Proposed by Jorge Tipe, Peru[/i]
2010 Contests, 3
[b](a)[/b]Prove that every pentagon with integral coordinates has at least two vertices , whose respective coordinates have the same parity.
[b](b)[/b]What is the smallest area possible of pentagons with integral coordinates.
Albanian National Mathematical Olympiad 2010---12 GRADE Question 3.
2007 Romania Team Selection Test, 4
Let $S$ be the set of $n$-uples $\left( x_{1}, x_{2}, \ldots, x_{n}\right)$ such that $x_{i}\in \{ 0, 1 \}$ for all $i \in \overline{1,n}$, where $n \geq 3$. Let $M(n)$ be the smallest integer with the property that any subset of $S$ with at least $M(n)$ elements contains at least three $n$-uples \[\left( x_{1}, \ldots, x_{n}\right), \, \left( y_{1}, \ldots, y_{n}\right), \, \left( z_{1}, \ldots, z_{n}\right) \] such that
\[\sum_{i=1}^{n}\left( x_{i}-y_{i}\right)^{2}= \sum_{i=1}^{n}\left( y_{i}-z_{i}\right)^{2}= \sum_{i=1}^{n}\left( z_{i}-x_{i}\right)^{2}. \]
(a) Prove that $M(n) \leq \left\lfloor \frac{2^{n+1}}{n}\right\rfloor+1$.
(b) Compute $M(3)$ and $M(4)$.
2006 Alexandru Myller, 3
$ 5 $ points are situated in the plane so that any three of them form a triangle of area at most $ 1. $ Prove that there is a trapezoid of area at most $ 3 $ which contains all these points ('including' here means that the points can also be on the sides of the trapezoid).